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m04 - Theorems on the Fourier Series

Module by: C. Sidney Burrus

Summary: A number of important theorems can be given on the Fourier series.

Theorems on the Fourier Series

Four of the most important theorems in the theory of Fourier analysis are the inversion theorem, the convolution theorem, the differentiation theorem, and Parseval's theorem [1] . All of these are based on the orthogonality of the basis function of the Fourier series and integral and all require knowledge of the convergence of the sums and integrals. The practical and theoretical use of Fourier analysis is greatly expanded if use is made of distributions or generalized functions [2] [3] . Because energy is an important measure of a function in signal processing applications, the Hilbert space of L 2 L 2 functions is a proper setting for the basic theory and a geometric view can be especially useful [4] [1] .

The following theorems and results concern the existence and convergence of the Fourier series and the discrete-time Fourier transform [5] . Details, discussions and proofs can be found in the cited references.

  • If f ( x ) f ( x ) has bounded variation in the interval ( − π , π ) ( − π , π ) , the Fourier series corresponding to f ( x ) f ( x ) converges to the value f ( x ) f ( x ) at any point within the interval, at which the function is continuous; it converges to the value 1 2 [ f ( x + 0 ) + f ( x − 0 ) ] 1 2 [ f ( x + 0 ) + f ( x − 0 ) ] at any such point at which the function is discontinuous. At the points π , − π π , − π it converges to the value 1 2 [ f ( − π + 0 ) + f ( π − 0 ) ] 1 2 [ f ( − π + 0 ) + f ( π − 0 ) ] . [6]
  • If f ( x ) f ( x ) is of bounded variation in ( − π , π ) ( − π , π ) , the Fourier series converges to f ( x ) f ( x ) , uniformly in any interval ( a , b ) ( a , b ) in which f ( x ) f ( x ) is continuous, the continuity at a a and b b being on both sides. [6]
  • If f ( x ) f ( x ) is of bounded variation in ( − π , π ) ( − π , π ) , the Fourier series converges to 1 2 [ f ( x + 0 ) + f ( x − 0 ) ] 1 2 [ f ( x + 0 ) + f ( x − 0 ) ] , bounded throughout the interval ( − π , π ) ( − π , π ) . [6]
  • If f ( x ) f ( x ) is bounded and if it is continuous in its domain at every point, with the exception of a finite number of points at which it may have ordinary discontinuities, and if the domain may be divided into a finite number of parts, such that in any one of them the function is monotone; or, in other words, the function has only a finite number of maxima and minima in its domain, the Fourier series of f ( x ) f ( x ) converges to f ( x ) f ( x ) at points of continuity and to 1 2 [ f ( x + 0 ) + f ( x − 0 ) ] 1 2 [ f ( x + 0 ) + f ( x − 0 ) ] at points of discontinuity. [6] [7]
  • If f ( x ) f ( x ) is such that, when the arbitrarily small neighborhoods of a finite number of points in whose neighborhood | f ( x ) | | f ( x ) | has no upper bound have been excluded, f ( x ) f ( x ) becomes a function with bounded variation, then the Fourier series converges to the value 1 2 [ f ( x + 0 ) + f ( x − 0 ) ] 1 2 [ f ( x + 0 ) + f ( x − 0 ) ] , at every point in ( − π , π ) ( − π , π ) , except the points of infinite discontinuity of the function, provided the improper integral ∫ − π π f ( x ) ⅆ x ∫ − π π f ( x ) ⅆ x exist, and is absolutely convergent. [6]
  • If f is of bounded variation, the Fourier series of f converges at every point x x to the value [ f ( x + 0 ) + f ( x − 0 ) ] / 2 [ f ( x + 0 ) + f ( x − 0 ) ] / 2 . If f is, in addition, continuous at every point of an interval I = ( a , b ) I = ( a , b ) , its Fourier series is uniformly convergent in I I . [8]
  • If a ( k ) a ( k ) and b ( k ) b ( k ) are absolutely summable, the Fourier series converges uniformly to f ( x ) f ( x ) which is continuous. [5]
  • If a ( k ) a ( k ) and b ( k ) b ( k ) are square summable, the Fourier series converges to f ( x ) f ( x ) where it is continuous, but not necessarily uniformly. [5]
  • Suppose that f ( x ) f ( x ) is periodic, of period X X , is defined and bounded on [ 0 , X ] [ 0 , X ] and that at least one of the following four conditions is satisfied: (i) f f is piecewise monotonic on [ 0 , X ] [ 0 , X ] , (ii) f f has a finite number of maxima and minima on [ 0 , X ] [ 0 , X ] and a finite number of discontinuities on [ 0 , X ] [ 0 , X ] , (iii) f f is of bounded variation on [ 0 , X ] [ 0 , X ] , (iv) f f is piecewise smooth on [ 0 , X ] [ 0 , X ] : then it will follow that the Fourier series coefficients may be defined through the defining integral, using proper Riemann integrals, and that the Fourier series converges to f ( x ) f ( x ) at a.a. x x , to f ( x ) f ( x ) at each point of continuity of f f , and to the value 1 2 [ f ( x − ) + f ( x + ) ] 1 2 [ f ( x − ) + f ( x + ) ] at all x x . [1]
  • For any 1 ≤ p < ∞ 1 ≤ p < ∞ and any f ∈ C p ( S 1 ) f ∈ C p ( S 1 ) , the partial sums
    S n = S n ( f ) = ∑ | k | ≤ n f ̂ ( k ) e k S n = S n ( f ) = ∑ | k | ≤ n f ̂ ( k ) e k (1)
    converge to f f , uniformly as n → ∞ n → ∞ ; in fact, || S n − f || ∞ || S n − f || ∞ is bounded by a constant multiple of n − p + 1 / 2 n − p + 1 / 2 . [4]

The Fourier series expansion results in transforming a periodic, continuous time function, x ˜ ( t ) x ˜ ( t ) , to two discrete indexed frequency functions, a ( k ) a ( k ) and b ( k ) b ( k ) that are not periodic.

References

  1. D. C. Champeney. (1987). A Handbook of Fourier Theorems. Cambridge: Cambridge University Press.
  2. A. Papoulis. (1962). The Fourier Integral and Its Applications. McGraw-Hill.
  3. R. N. Bracewell. (1985). The Fourier Transform and Its Applications. (Third). New York: McGraw-Hill.
  4. H. Dym and H. P. McKean. (1972). Fourier Series and Integrals. New York: Academic Press.
  5. A. V. Oppenheim and R. W. Schafer. (1989). Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall.
  6. E. W. Hobson. (1926). The Theory of Functions of a Real Variable and the Theory of Fourier's Series. (Second, Vol. 2). New York: Dover.
  7. H. S. Carslaw. (1906, 1930). Theory of Fourier's Series and Integrals. (third). New York: Dover.
  8. A. Zygmund. (1935, 1955). Trigonometrical Series. New York: Dover.

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